## Wednesday, January 23, 2008

### Light Deflection at the Sun

Albert Einstein shot to fame in November 1919 when an announcement was made by British astronomers at a scientific meeting in London: Measurements of starlight during a solar eclipse earlier that year had vindicated the gravitational deflection of light at the Sun, as predicted by General Relativity. Moreover, it was argued, the data were not compatible with Newton's time-honoured theory of gravitation.

In Einstein's theory of General Relativity, just four years old in November 1919, space-time in the vicinity of a large mass such as the Sun is curved, and light rays passing nearby the Sun get bent. As a consequence, the actual location of a star observed in the vicinity of the Sun ("wahre Position") is not at the position where it appears to be ("scheinbare Position"), but a bit closer to the Sun. The deflection angle δ for the light passing just at the rim of the Sun can be calculated from the speed of light, c, the gravitational constant, G, and the mass and radius of the Sun. It is

- we will come back to the extra factor (1+γ)/2 in a second. If wee keep in mind that the Sun has an apparent diameter of half a degree, or 1800 arc seconds, the deflection of light from a star close to the rim of the Sun is just 1/1000 of the Sun's diameter. Moreover, for light passing at larger distances d from the Sun, the angle drops as 1/d - so this is a tiny effect!

Actually, a similar deflection of light had already been calculated before Einstein. The British physicist Henry Cavendish, and, most notably, the German astronomer Johann Georg von Soldner had used Newton's mechanics and calculated the hyperbolic trajectory of a particle which passes at the speed of light nearby a large mass. This calculation yields a deflection angle that is just half as big as the value obtained from General Relativity.

This difference between the two calculations is nowadays encoded in a parameter called γ, where γ = 1, or (1+γ)/2 = 1, corresponds to the bending of light as predicted by General Relativity, while γ = 0, or (1+γ)/2 = 1/2, is the value for the Newtonian calculation. Actually, γ is just one out of a set of several parameters which are used in a framework called parametrised post-Newtonian formalism. This formalism had been developed to describe, in a unified framework, the observable consequences of different possible theories of gravitation. For example, γ describes how much space curvature is produced by a unit rest mass. Newtonian gravity comes with γ = 0 (no curvature), and General Relativity has &gamma = 1. Other conceivable theories of gravitation might come with still other values of γ, and the measurement of γ is a way to distinguish between them.

Negative of the solar eclipse of May 29, 1919, photographed by Andrew Crommelin in Sobral, Brazil. Stars are marked by horizontal lines. (via Wikipedia, from F. W. Dyson, A. S. Eddington, and C. Davidson, Phil. Trans. Royal Soc. London. Series A 220 (1920) 291-333, page 332)
While Soldner had apologised to the readers of his 1801 paper for calculating an effect that he judged unobservable, astronomers a hundred years later had more confidence in their capabilities. Since it is not possible to observe stars in the vicinity of the Sun under normal circumstances, they had to seize the rare opportunity of a total eclipse of the Sun, when stars nearby are visible. By comparing the apparent positions of these stars to the true positions (measured at night, at a different time of the year, when the effect of gravitational bending by the Sun can be neglected), the deflection of light by the gravitational field of the Sun could be established.

Motivated by these considerations, the British astronomer and relativity aficionado Arthur Stanley Eddington organised two expeditions to observe a solar eclipse on May 29, 1919, with a zone of totality roughly along the equator. He travelled to Principe, an island in the Atlantic ocean, while a second team observed the event from Sobral in Brazil.

The results of these observations were made public at the meeting in London in November 1919 that made Einstein a scientific star: The measured deflection of light did fit to the Einstein value, while it was much less compatible with the Newtonian bending.

Of course, not only Einstein denialists point out the huge error bars of the eclipse measurements. Eddington had only a few stars on his photographic plates, due to bad weather, and the main telescope of the Sobral team had suffered misalignment caused by heating in the plain daylight before the eclipse. As a result, data taken with this instrument had been discarded - which is a tricky point, since they seem to have favoured a Newtonian value for the light deflection.

However, the 1919 eclipse data have just been the beginning of a long series of ever-improving measurements of the gravitational deflection of light. This finally brings us to our plottl.

Its upper part shows, as a function of time along the horizontal axis, the results for measurements of the light-bending parameter (1+γ)/2, as established by different methods.

Improvement of the measurement of the gravitational bending of light and radio waves by the Sun (upper part of the figure) over the last 80 years. The horizontal black line at (1+γ)/2 = 1 corresponds to the prediction of General Relativity. [Source: The Confrontation between General Relativity and Experiment by Clifford Will, Living Rev. Relativity 9 (2006), cited on <2008/01/13> (http://www.livingreviews.org/lrr-2006-3), Figure 5.]

Marked in red are data from measurements with visible light, all but one taken at solar eclipses. The 1919 eclipse is the left-most data point. As we know already, these eclipse data come with large uncertainties and huge error bars - some do not even fit on the plot (the red arrows at the upper edge) - and there has been been only a modest progress up to the 1970s. The last eclipse expedition with published data about light deflection was to Mauritania, resulting in the paper Gravitational deflection of light: Solar eclipse of 30 June 1973. I. Description of procedures and final result. by Brune et al., Astron. J. 81 (1976) 452-454.

However, since the advent of satellites, it is not necessary anymore to wait for an eclipse to observe stars in the vicinity of the Sun. Thus, an analysis of the star catalogue established by the astrometry satellite Hipparcos could confirm the Einstein value for the bending of light, (1+γ)/2 = 1, to within 0.1 percent (FroeschlĂ©, M., Mignard, F., and Arenou, F.: Determination of the PPN parameter γ with the Hipparcos data, Proceedings from the Hipparcos Venice ’97 Symposium (ESA, Noordwijk, Netherlands, 1997) - PDF)

And, of course, light is only part of the electromagnetic spectrum. For example, one can use radio telescopes to measure the deflection of radio signals from quasars, completely analogous to the the measurement of starlight, but with the bonus that there is no need to wait for eclipses. Early attempts to do so from the 1960s and 1970s are marked by the blue dots - the results vindicate, and improve on, the optical observations (E. B. Fomalont, R. A. Sramek: Measurements of the Solar Gravitational Deflection of Radio Waves in Agreement with General Relativity, Phys. Rev. Lett. 36 (1976) 1475-1478.)

And finally, the interferometric combination of radio telescopes from all over the globe has further improved the quasar data. These so called VLBI (Very Long Baseline Interferometry) light deflection measurements have reached an accuracy of 0.02 percent, and they fit perfectly well to the predictions of General Relativity (Shapiro, S.S., Davis, J.L., Lebach, D.E., and Gregory, J.S.: Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1979-1999, Phys. Rev. Lett. 92 (2004) 121101.)

Thus, Eddington had got it right, and as it looks from today's data, General Relativity rules!

PS: The lower part of the plot shows the so far best determination of γ. It uses a different but related effect, the so-called Shapiro time delay, which is based on the apparently reduced speed of light in the vicinity of large masses. This time delay can now be measured extremely precisely thanks to the telemetry data of spacecraft travelling around in the Solar System - Viking, Voyager, Cassini. The Shapiro time-delay measurements using the Cassini spacecraft yielded an agreement with General relativity to 0.001 percent (B. Bertotti, L. Iess and P. Tortora: A test of general relativity using radio links with the Cassini spacecraft, Nature 425 (2003) 374-376).

Einstein Online has a great first introduction to the Gravitational deflection of light by Steven and Irwin Shapiro.

For the calculations of Cavendish and Soldner, see Clifford M. Will: Henry Cavendish, Johann von Soldner, and the deflection of light, American Journal of Physics 56 (1988) 413-415 (subscription required).

The 1919 eclipse expedition and its motivation and background by Einstein's prediction of the bending of light is described, e.g., by Peter Coles: Einstein, Eddington and the 1919 Eclipse, arXiv:astro-ph/0102462v1. For a recent discussion about the analysis of the photographic data, see Daniel Kennefick: Not Only Because of Theory: Dyson, Eddington and the Competing Myths of the 1919 Eclipse Expedition, arXiv:0709.0685v2 [physics.hist-ph]. The original paper of the 1919 eclipse expedition is F. W. Dyson, A. S. Eddington, and C. Davidson: A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919, Philosophical Transactions of the Royal Society of London. Series A 222 (1920) 291-333.

This post is a latecomer to our A Plottl A Day series.

Neil' said...

Yes, these were famous and ground-breaking experiments even under the eventual criticism that accuracy was poor and the appearance of confirmation was perhaps somewhat lucky a correlation. It turns out that if Einstein could have done the experiment earlier (I forget what held it up and when) he would have been embarrassed, since his first prediction was for half the final value (since IIRC he didn't take true space curvature, the "dimple", around the sun into account.) BTW, people still argue over whether space-time is "really curved" or that's just a way to talk about the distortion of metric standards, or maybe it's two ways to talk about the same thing? (reminds of the interpretation of "collapse of the wave function?)

I am also interested in comparing to the issue of deflection of high-speed matter, also the question of fields produced by extended planar masses (no gravity "dimple") not just concentrated centers of mass. I heard an interesting claim on a big thread in Cosmic Variance about the field of a large planar mass PM, also I think applicable to that of a compact body (but not to the "Rindler field", which is alleged not to be just like that of a real PM.) Commenter G.E. said that a mass zipping parallel to PM (as well as near a compact mass, but harder to define) would have higher acceleration (defined in rest observer’s simple fashion, z dot dot with z perp. to the PM., details if requested.) I thought that was weird since the equivalence principle says, that the field should be like in a giant accelerating elevator - in which case the side-moving mass must hit the floor at the same time as one dropped straight down. I got the impression that this higher acceleration was due to the "dimple" distortion for a compact mass, analogous to why light deflected twice as much as simple attraction in flat space, but I didn't get why it would also apply to the parallel field of a planar mass sheet.

Sure, we can explain a higher force on the moving particle's increased "relativistic mass", but the transformations for force and acceleration should all cancel out and bring rest-frame tangential acceleration back to the same value as a simple drop, as the "elevator" implies.

But to make it more confusing, G.E. then seemed at least to contradict himself and say, that this differential acceleration really didn’t violate the Equivalence Principle despite the intuitive notion of comparing those differential accelerations (it seems to me that even in a small region, different accelerations could not be transformed away in tandem - the falling observer would know he was not in true inertial space.) G.E. provided what seemed competent but rather sophistic reasoning and tricky metaphors to support his counterintuitive claims. What’s anyone's knowledge/opinion on this? tx

Phil Warnell said...

Hi Bee & Stefan,

Great post, I particularly liked how you spelled out that the accuracy of measurement in relation to the predicted values was a tricky thing for some time. This in turn allowed for much skepticism and debate for many years and was partially responsible for why Einstein was not awarded the Nobel Prize in direct relation to it. I would also expect that many not familiar with the subject are unaware that Newtonian gravity also predicts a deflection, yet as you have explained, just not as severe.

Of course one could claim that this would only be true if light is considered as particle phenomena and not that of a wave. The beauty of course is that in Einstein’s treatment this doesn’t apply for what is perceived as a defection is a direct result of the shape of the related space/time anything travels through. It may appear curved, yet with the concept of a straight line being the shortest distance between two points, this is the path taken (fifth postulate denied). Therefore, whether particle or wave, a straight line is still a straight line.

This of course renders gravity to be not a force at all, yet rather simply the logical consequence of architecture or geometry if you prefer. It is of little wonder that it stubbornly refuses to mess with quantum theory for as this demands that it be a phenomena resultant of a force and not simply a path determined by shape. For many years Einstein believed that quantum theory would yield before gravity and yet today primarily the reverse is suspected to be true. I have often wondered if I will see the day when it is decided which is correct. Then again, neither may be and an entirely new concept could prove to be the answer. So Bee, this is your quest. It is a noble one and yet it would be something many would be reluctant to tackle, although many have tried and continue to. My only comfort would be to relate to Einstein when he said:

“ It is open to every man to choose the direction of his striving: and also every man may draw from Lessing’s fine saying, that the search for truth is more precious than its possession. “

Of course today “man” would be replaced with “person” for in 1940 the term would have been meant as not to distinguish gender, in the context described.

So good hunting and keep the coffee hot :-)

Best,

Phil

Bee said...

Oops, I didn't fully realize you used the picture from my thesis. I could have given you an English version!

Ned Wright said...

Einstein "knew" Newton was wrong because an inverse square law couldn't fit into special relativity, and in 1913 Nordstrom made relativistic gravity theory that predicted zero deflection of light. But the "Einstein elevator" thought experiment leads to a result that matches Newton, so Einstein
wrote
to George Ellery Hale in 1913 asking him to measure the deflection, with a prediction of 0.84". The final version of GR in 1915 doubled the prediction.

Uncle Al said...

http://arXiv.org/abs/gr-qc/9909014

Commentary on "light falls with twice the acceleration of ordinary matter"

stefan said...

Dear Bee,

I didn't fully realize you used the picture from my thesis. I could have given you an English version!

That figure was the first to cross my mind when looking for an illustration - sorry for omitting the proper credits ;-). Concerning the German text, well, I thought about asking you for a version with English labels. But then, I thought, Sonne and Erde is not too different from Sun and Earth and should be clear, and I actually like the beautiful German words "scheinbar" and "wahr". So, I decided to keep the figure with the German text - and to see it as a tribute to Soldner and Einstein...

Best, Stefan

stefan said...

Hi Ned, hi all,

thank you very much for the link to the Einstein letter to Hale, and for all the additional remarks about the pre-GR light deflection!

We didn't want to add these extra twists to the post, since it seemed to be too long already anyway ;-)

But true, these are important points!

I wonder, for example, Maxwell's theory of light and the special theory of relativity had been accepted by physicists by 1919. So, it should have been clear at the time that Soldner's Newtonian derivation of a deflection of light was based on quite shaky physics reasoning. Thus, why should one have expected this Newtonian deflection to occur in the first place? As Ned has pointed out, the most straight-forward combination of gravitation and special relativity by NordstrĂ¶m gives no light deflection at all.

So (I have read that somewhere, not sure where), Eddington may have deliberately framed the experiment as "deciding" between a "Newton deflection" and a "Einstein deflection" to make it appear more interesting?

And it's a curious twist that the Prague beta-version of General Relativity actually yields the "Newtonian" deflection angle, and half the final GR value!

BTW, while the correspondence of Einstein with Hale about possible measurements of the deflection of light was blocked by the war, Einstein had been in contact about possible astronomical tests of GR with astronomer Erwin Freundlich at Postdam since his Prague days. Freundlich was keen to check the astronomical predictions of General Relativity, and embarked on an expedition to observe the August 21, 1914 ecplise in Crimea. He was indeed hoping to see half the full GR value, back at that time.

Unfortunately, the First Worl War broke out while he was travelling through Russia, and he was arrested as a hostile alien and a putative spy - he was carrying along telescopes, after all. He was allowed to return to Germany later, but without his instruments.

Which brings me to Phils remark concerning the large errors in the light deflection measurements...

This in turn allowed for much skepticism and debate for many years and was partially responsible for why Einstein was not awarded the Nobel Prize in direct relation to it.

Actually, it seems that most initial scepticism against GR (after 1919, now referring to the full version) was aired because of the total failure to measure the predicted redshift of light emitted from the Sun. In Potsdam, there was even build a dedicated new solar observatory, the Einsteinturm with Freundlich as director, specially to measure the solar redshift, but without success. Gravitational redshift could be established for the first time only in 1925 with light from Sirius B, a white dwarf star, but not for the Sun for a very long time.

If you want to read more about this, there is a series of papers by science historian Klaus Hentschel about tbis (unfortunately, subscription reqired), and a recent book by Jeff Crelinsten (which I have not read, so I can't comment).

Best, Stefan

Coin said...

I'm finding myself heavily distracted by the NYT article about the results of the Eddington expedition. Two things jump out at me: One, although they don't seem to have gotten everything right, I'm amazed how specific and clear their reporting on the experimental results of the expedition-- giving exact degrees of deflection and error bars-- was. Would the New York Times EVER provide that amount of detail about the results of a scientific experiment today?

Two, this is likely just my inexperience with GR at work, but I don't understand what this part is going on about:

Two of the consequences of Einstein's theory, he continued, namely, the motion of Mercury's perhelion and the bending of light by gravitation, might now be looked on as established, "at least with great probability". There was, however, a third predicted consequence, which was a 'shift of the lines in the spectrum toward the red in a strong gravitational field. The effect in the solar spectrum would amount to one-twentieth of the Angstrom unit, the same as that due to a motion of one-half kilometer per second away from the sun. Dr. St. John had looked for this effect without success. If this failure were taken as final it would mean that parts of Einstein's theory would need revision, but the parts already verified would remain.

...what?

Plato said...

Just out of curiosity can the letter of Einstein's letter to the American G.E. Hale. be translate verbatim?

Also I think the point about the shift to non-euclidean geometries by Riemann through Grossman was to have some introduction for Einstein to have ever considered the value of that experiment?

There was a "geometrical tradition" there from Saccherri to Lobachevsky, Bolyai and Gauss, and finally to Riemann.

Helmholtz was the first to advocate the idea that human beings, living in a non-Euclidean world, would develop an ability of visualization which would make them regard the laws of non-Euclidean geometry as necessary and self-evident, in the same fashion as the laws of Euclidean geometry appear self-evident to us

There is a "deductive geometrical feature" of those things self evident. That's the point of the geometry as a "basis of the logic" behind that experiment.

Phil Warnell said...

Stefan,

“I wonder, for example, Maxwell's theory of light and the special theory of relativity had been accepted by physicists by 1919. So, it should have been clear at the time that Soldner's Newtonian derivation of a deflection of light was based on quite shaky physics reasoning.”

I’m confused here, for are you saying that Newtonian Mechanic’s would yield no deviation or are you saying that from the Maxwell perspective of light as being a wave coupled with SR that this would not be so. From the Newtonian perspective light was a particle phenomenon (have mass) which would be affected by classical gravity regardless of SR implications. What I consider the main issue here is that with Newtonian gravity a particle would be deflected while a wave would not for a wave would not have mass in the Newtonian perspective. In GR however it doesn’t matter since we are not talking about deflection resultant of a force as Newton would consider but rather a path dictated by the architectural mandate of space/time. Which is not a force yet rather holds the appearance of a force.

“Actually, it seems that most initial scepticism against GR (after 1919, now referring to the full version) was aired because of the total failure to measure the predicted redshift of light emitted from the Sun.”

This is very interesting for I wasn’t aware of this. I’m also surprised that they felt they were technically capable to detect such a miniscule deviation at the time. I’m glad they weren’t waiting to detect gravity waves before conceding, for we would still be waiting:-) As an related aside question, have they ever conducted a experiment that would confirm that gravity’s influence in terms of propagation is limited to c or is it only assumed that because other things are consistent with the theory it need not be confirmed?

Regards,

Phil

Phil Warnell said...

Hi Uncle,
I like this paper you sited (http://arXiv.org/abs/gr-qc/9909014) for it concludes the following:

“We can thus tell our students with confidence that kinetic energy has weight, not just as a theoretical expectation, but as an experimental fact.”

I do however think that this is perhaps still the wrong way to state the conclusion. For the conclusion should be that no matter what is considered to be travelling through any one region of space/time that it will be limited to be only able to follow the path presented in relation to the phenomena observed. I think to mix in the weight (mass) consideration is a red herring and still leaves room for ambiguity and confusion.

Regards,

Phil

stefan said...

Hi Plato,

... can Einstein's letter to the American G.E. Hale be translated verbatim?

Here it is (my translation):
___________

Zurich, 14 October 1913

Highly esteemed colleague,

a simple theoretical consideration makes it plausible to assume that light rays will experience a deviation in a gravitational field.

[Grav. field] [Light ray]

At the rim of the Sun, this deflection should amount to 0.84" and decrease as 1/R (R = [strike]Sonnenradius[/strike] distance from the centre of the Sun).

[Earth] [Sun]

Thus, it would be of utter interest to know up to which proximity to the Sun bright fixed stars can be seen using the strongest magnification in plain daylight (without eclipse).

_________________

stefan said...

Hi coin,

that part of the NYT article discusses the point I've mentioned before, the gravitational redshift.

In a nutshell, the frequency of light that is escaping from the gravitational well of a large mass is shifted towards lower frequencies / longer wavelengths - the photon looses energy. This is a consequence of the equivalence principle, and holds in particular for General Relativity.

Astronomers thought they should be able to detect a slight redshift of the Fraunhofer lines of the Solar spectrum with respect to the lines as measured in the lab. However, as Mount Wilson astronomer Charles E. St. John (the guy mentioned in the article) writes in 1917 in the paper "A Search for an Einstein Relativity-Gravitational Effect in the Sun" (PNAS 1917 3: 450-452):

"The general conclusion from the investigation is that within the limits of error the measurements show no evidence of an effect of the order deduced from the equivalence relativity principle."

So, St.John was quite skeptical when he heard about the the ecplise data, which did not fit to his own measurements.

As far as I know, the problem is turbulence and radial motion in the Sun's upper layers, which blurs the gravitational redshift. It was seen in other stars, and then, in 1959, in the laboratory for γ rays via the MĂ¶ssbauer effect - that's the famous experiment by Rebka, Pound, and Snyder.

To counter the gravitational redshift (or blueshift, if light or radio waves come in from a higher gravitational potential), the atomic clocks of the GPS satellites are slightly detuned with respect to the atomic clocks on the Earth's surface.

Best, Stefan

stefan said...

Hi Phil,

I think you are not confused at all ;-)... What I wanted to say is exactly as you say: from the Newtonian perspective, Soldners calculation is perfectly right - but, as was well-established by 1919, the Newtonian perspective is not applicable to light, which is an electromagnetic wave and subject to special relativity... and for Maxwell/SRT coupled to gravity à la NordstrĂ¶m, there is no deflection.

As a related aside question, have they ever conducted a experiment that would confirm that gravity’s influence in terms of propagation is limited to c?

You mean, experiments to check whether changes in the gravitational field propagate with the speed of light? You may google for the Fomalont Kopeikin experiment (the same Fomalont mentioned in the post from the early quasar deflection measurements). They measured the deflection of the light of a quasar by Jupiter, and argued that they could deduce a speed of gravity within (1.06 ± 0.21) c. (arXiv:astro-ph/0302294).

But there was quite a controversy about the interpretation of their measurements (not about the data), and whether it actually did say anything about the speed of gravity. Unfortunatley, I do not know what was the last word in this issue, if there has been one so far.

Best, Stefan

Phil Warnell said...

Hi Stefan,

“I think you are not confused at all ;-)”

Whew! Thanks Stefan, I thought it might have become time to give up physics as my hobby and take up knitting or something:-)

“They measured the deflection of the light of a quasar by Jupiter, and argued that they could deduce a speed of gravity within (1.06 ± 0.21) c. (arXiv:astro-ph/0302294)……….. But there was quite a controversy about the interpretation of their measurements………..”

Yes now that you point it out I vaguely recall hearing something about this. From what I can so far gather it’s somewhat still up in the air. I guess we will have to wait until they detect gravity waves themselves that can be correlated with some major visible gravity event like two galactic super massive black holes colliding or some such. Strangely enough although GR insists that gravity is limited to light speed there are other elements involved that suggest some sort of non-local nature as well. This is something that apparently haunted Einstein later in life. I believe I read Smolin comment on it at one point. In some ways it would almost be a blessing if gravity had some non dismissible non-local aspect for that could draw it closer in being meshed with QM. I’ve also read that some of the (large) extra dimensional brane theories would have an easier time if this were true.

Regards,

Phil

Plato said...

There is always a difference between the basis of "discrete and continuous" and this seems to reflect itself in the way "branches of science" are interpreted?

One from a "gravitational interpreted photon" or, "one from a topological basis? :)

Gaussian Coordinates

We can sum this up as follows: Gauss invented a method for the mathematical treatment of continua in general, in which ?size-relations? (?distances? between neighbouring points) are defined. To every point of a continuum are assigned as many numbers (Gaussian co-ordinates) as the continuum has dimensions. This is done in such a way, that only one meaning can be attached to the assignment, and that numbers (Gaussian co-ordinates) which differ by an indefinitely small amount are assigned to adjacent points. The Gaussian co-ordinate system is a logical generalisation of the Cartesian co-ordinate system. It is also applicable to non-Euclidean continua, but only when, with respect to the defined ?size? or ? distance,? small parts of the continuum under consideration behave more nearly like a Euclidean system, the smaller the part of the continuum under our notice.

So one has to find some supporting historical documents for the way they believe?:)

Dr. Sanford Aranoff said...

What happens as Rs goes to zero - will delta go to 2 pi? Delta near a neutron star is 100°. What is delta near a black hole? Deflection so large that the light never passes the black hole?

stefan said...

Hi Sanford,

hm, I would have to look up the limits of validity of the simple deflection formula above.

I am quite sure that it can not be applied anymore to light rays passing nearby the horizon of a Schwarzschild black hole. But in that case, there have been simulations using ray tracing in the Schwarzschild metric - see for example this PDF.

Light rays can orbit around the horizon once or even twice. An accretion disc would look "bent up" - you might have seen the beautiful illustration on the title page of the book by Begelman and Rees on Black Holes. For the same reasons, such disks would show a very characteristic pattern that astronomers hope to be able to detect in the not too far future.

Hope that helps a bit...
Cheers, Stefan